Milliken's tree theorem

In mathematics, Milliken's tree theorem in combinatorics is a partition theorem generalizing Ramsey's theorem to infinite trees, objects with more structure than sets.

Let T be a finitely splitting rooted tree of height ω, n a positive integer, and ${\displaystyle \mathbb {S} _{T}^{n}}$ the collection of all strongly embedded subtrees of T of height n. In one of its simple forms, Milliken's tree theorem states that if ${\displaystyle \mathbb {S} _{T}^{n}=C_{1}\cup ...\cup C_{r}}$ then for some strongly embedded infinite subtree R of T, ${\displaystyle \mathbb {S} _{R}^{n}\subset C_{i}}$ for some i ≤ r.

This immediately implies Ramsey's theorem; take the tree T to be a linear ordering on ω vertices.

Define ${\displaystyle \mathbb {S} ^{n}=\bigcup _{T}\mathbb {S} _{T}^{n}}$ where T ranges over finitely splitting rooted trees of height ω. Milliken's tree theorem says that not only is ${\displaystyle \mathbb {S} ^{n}}$ partition regular for each n < ω, but that the homogeneous subtree R guaranteed by the theorem is strongly embedded in T.

Strong embedding

Call T an α-tree if each branch of T has cardinality α. Define Succ(p, P)= ${\displaystyle \{q\in P:q\geq p\}}$, and ${\displaystyle IS(p,P)}$ to be the set of immediate successors of p in P. Suppose S is an α-tree and T is a β-tree, with 0 ≤ α ≤ β ≤ ω. S is strongly embedded in T if:

• ${\displaystyle S\subset T}$, and the partial order on S is induced from T,
• if ${\displaystyle s\in S}$ is nonmaximal in S and ${\displaystyle t\in IS(s,T)}$, then ${\displaystyle |Succ(t,T)\cap IS(s,S)|=1}$,
• there exists a strictly increasing function from ${\displaystyle \alpha }$ to ${\displaystyle \beta }$, such that ${\displaystyle S(n)\subset T(f(n)).}$

Intuitively, for S to be strongly embedded in T,

• S must be a subset of T with the induced partial order
• S must preserve the branching structure of T; i.e., if a nonmaximal node in S has n immediate successors in T, then it has n immediate successors in S
• S preserves the level structure of T; all nodes on a common level of S must be on a common level in T.

References

1. Keith R. Milliken, A Ramsey Theorem for Trees J. Comb. Theory (Series A) 26 (1979), 215-237
2. Keith R. Milliken, A Partition Theorem for the Infinite Subtrees of a Tree, Trans. Amer. Math. Soc. 263 No.1 (1981), 137-148.